When taking photographs of a dim-lit scene, you could think that choosing the smallest ISO setting on your camera will help with reducing the overall noise of the picture. What matters for real however, is the total number of counts that the sensor has recorded. You cannot change how much light will reach the sensor for a given exposure time and aperture setting, but you can change the way the sensor will count them. This is done precisely by the ISO setting, and finding the range where the handling of the signal is best is an easy procedure.

Finding the best ISO for astrophotography on your camera

Find yourself a nice dark scene. I personally chose a book shelf just before sunset. Set your camera on a tripod (or on any appropriate substitute for a tripod), focus it on the scene. Maybe choose an aperture value that will give you a sharp image. By this, I mean do not shoot wide open because the defects of the optics might play a bigger role in this case. Also, do not forget to shoot RAW as Jpegs would not retain enough information for this to work. Finally, I went with the following steps:

Go the the maximum allowed native ISO value (on my camera: 12800).

Expose the scene correctly at this value using the exposure time and aperture settings. Take the picture.

Decrease the ISO value by one stop, take the picture again. (It should be twice as dark)

Repeat the above step until you reach the base ISO of your camera, usually 100.

You should be left with a bunch of RAW files becoming gradually darker. Import them into your favorite RAW editing software (I use darktable). Deactivate any noise reduction algorithm that might kick in automatically. For now the photographs should only be debayered, without any added correction. Now you can start bringing the exposure of the pictures to the same value: increase the exposure value by adding one stop to the picture taken at the second ISO, two stops to the one taken at the thrid ISO, etc.
I was left with the following situation:

ISO

Aperture

exposure time

exposure correction

12800

f/3.5

1/10

0

6400

f/3.5

1/10

+1

3200

f/3.5

1/10

+2

1600

f/3.5

1/10

+3

800

f/3.5

1/10

+4

400

f/3.5

1/10

+5

200

f/3.5

1/10

+6

100

f/3.5

1/10

+7

Now you should have a set of pictures with equal brightnesses. Here is what I got for my Nikon D750. (You can open the picture in a new tab to obtain the full resolution.)
Clearly, ISO 100 does not give the best result here. I would personally go with ISO 6400, it seems to be preserving the details in the picture the best. However at the end of the day, the differences are not extremely striking. It makes sense as approximately the same number of photons have hit the sensor for each of the photographs above; independently of how the electronics inside the camera handled the incoming signal.